Coupier, David; Hirsch, Christian
(2018):
Coalescence of Euclidean geodesics on the PoissonDelaunay triangulation.
In: Bernoulli, Vol. 24, No. 4A: pp. 27212751

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Abstract
Let us consider Euclidean firstpassage percolation on the PoissonDelaunay triangulation. We prove almost sure coalescence of any two semiinfinite geodesics with the same asymptotic direction. The proof is based on an argument of BurtonKeane type and makes use of the concentration property for shortestpath lengths in the considered graphs. Moreover, by considering the specific example of the relative neighborhood graph, we illustrate that our approach extends to further wellknown graphs in computational geometry. As an application, we show that the expected number of semiinfinite geodesics starting at a given vertex and leaving a disk of a certain radius grows at most sublinearly in the radius.